TEACHER NOTES MATH NSPIRED
|
|
- Jewel Kelly
- 5 years ago
- Views:
Transcription
1 Math Objectives Students will produce various graphs of Taylor polynomials. Students will discover how the accuracy of a Taylor polynomial is associated with the degree of the Taylor polynomial. Students will visualize the accuracy and relate this to symmetry, arc length, and a point of discontinuity. Students will eamine certain common functions and draw specific conclusions about the Taylor polynomial associated with these functions. Students will look for and make use of structure. (CCSS Mathematical Practice) Students will reason abstractly and quantitatively. (CCSS Mathematical Practice) TI-Nspire Technology Skills: Download a TI-Nspire document Open a document Move between pages Manipulate a slider Vocabulary Taylor polynomial degree n accuracy of the approimation neighborhood of a About the Lesson This lesson involves Taylor polynomials associated with five common functions. As a result, students will: Learn about Taylor polynomials graphically and numerically. Conjecture about factors that affect the accuracy of Taylor polynomial approimations. Visualize the effects of n and a on the accuracy of the approimation. TI-Nspire Navigator System Use Screen Capture to demonstrate various approimations depending on the degree and a (page 2.3). Use Teacher Edition computer software to illustrate various Taylor polynomials. Encourage students to try several different values of n and a. Tech Tips: Make sure the font size on your TI-Nspire handheld is set to Medium. In Graphs, you can view the function entry line by pressing /G, and then enter a function. Press / ~ and select Lists & Spreadsheets to insert a new Lists & Spreadsheets page. Lesson Materials: Student Activity Taylor_Polynomial_Eamples.pdf Taylor_Polynomial_Eamples.doc TI-Nspire document Taylor_Polynomial_Eamples.tns Visit for lesson updates and tech tip videos. 20 Teas Instruments Incorporated education.ti.com
2 Use Screen Capture to display Taylor polynomials associated with y = sin in order to discover the symmetry and the changes in the polynomials as n increases by. At the end of this activity, ask students to consider other common functions and the associated Taylor polynomials. Discussion Points and Possible Answers Tech Tip: If you use the slider to move away from a = 0 and then back to a = 0, occasionally the calculator will display a very small number, not eactly 0. There may be several reasons for this approimation that can lead to interesting discussions. Ask students to determine the increment in the eamples involving y sin and y cos, and why this increment was selected. Move to page In the first eample, the graph of y e is dotted and the graph of the Taylor polynomial of degree n at a is solid. Use the slider arrows to change the degree, n, or the value of a. a. With a = 0, set n = Graph the first degree Taylor polynomial, T, at 0. Describe the graph of y T ( ). Answer: The graph of the first degree Taylor polynomial is a straight line, the tangent line to the graph of y e at the point (0,). b. Use the graph of y T ( ) and the Trace All feature to describe the accuracy of the Taylor polynomial approimation as moves farther from a = 0. Answer: As moves farther from a = 0, the approimation becomes less accurate. This is confirmed by visual inspection and numerically using the Trace All feature. 20 Teas Instruments Incorporated 2 education.ti.com
3 c. Set n = 2. Describe the graph of y T ( ) 2, the second degree polynomial at 0. Answer: The graph of the second degree Taylor polynomial is a parabola. d. Set n = 3. Describe the graph of y T ( ) 3, the third degree polynomial at 0. Answer: The graph of the third degree Taylor polynomial 3 looks like the graph of y, a cubic polynomial, shifted vertically up unit. e. Consider the graph of other Taylor polynomials for n 4. Describe the accuracy of the Taylor polynomial approimation as n increases. Answer: As n increases, the Taylor polynomial approimation becomes more accurate. For larger values of n, the graph of the Taylor polynomial seems to lie more closely on the graph of y e, especially to the left of a = 0. Move to page 2.3. On this Lists & Spreadsheets page you may enter values for in column A. The following values will be computed automatically: f( ), taylorf( ), and f( ) taylorf ( ), columns B, C, and D respectively. These resulting values are dependent upon the current values of n and a. 2. Adjust the values of n and a on page 2.2 as necessary and use the Lists & Spreadsheets page to answer the following questions. a. For a fied value of n, describe the accuracy of the Taylor polynomial approimation as the values of are farther away from a. Answer: As moves farther away from a, the approimation is worse. The closer is to a, the more accurate the approimation given by the Taylor polynomial. b. For fied values of a and, describe the accuracy of the Taylor polynomial approimation as n increases. Answer: For fied values of a and, as the degree of the Taylor polynomial increases, that is, as n increases, the approimation given by the Taylor polynomial becomes more accurate. 20 Teas Instruments Incorporated 3 education.ti.com
4 Move to page In this eample, the graph of y ln( ) is dotted and the graph of the Taylor polynomial of degree n at a is solid. Use the slider arrows to change the degree, n, or the value of a. Adjust the values of n and a as necessary to answer the following questions. a. For a = 2, describe the accuracy of the Taylor polynomial approimation as n increases. Answer: As n increases, the Taylor approimation becomes more accurate. As n increases, to the left of a = 2, the graph of the Taylor polynomial lies closer to the graph of y ln and always decreases without bound as 0. As n increases, to the right of a = 2, the graph of the Taylor polynomial appears to be a very good approimation up to approimately = 4. b. Describe the behavior of each Taylor polynomial as and as. What happens to the graph of the Taylor polynomial, as, as n increases by, for eample, from n = 6 to n = 7? Eplain why this behavior alternates as n increases. Answer: The right tail of the graph of the Taylor polynomial alternates: for n even, as, taylorf, and for n odd, as, taylorf. As n increases by, an additional term ( n f ) (2) is added to the Taylor polynomial, of the form ( 2) n. This term dominates, or controls, n! the behavior of the polynomial for large values of. Therefore, as increases, ( 2) n becomes large positive, and the behavior of the Taylor polynomial is controlled by the sign of the nth derivative at 2. This derivative term alternates in sign, causing the graph of the Taylor polynomial to alternate as increases. c. For a = 0.3, consider various Taylor polynomials of different degrees. Eplain why the Taylor polynomial appears to be a very good approimation to the left of a = 0.3 but diverges rapidly to the right of a = 0.3. Answer: Although it will become clearer in the net eample, the Taylor approimation is accurate on a symmetric interval about the point a. In this eample, for a = 0.3, the Taylor polynomial can only provide an accurate approimation up to (but not including) = 0.6. This is because the function f ( ) ln has domain 0. Teacher Tip: Ask students to adjust the Window Settings in order to more closely observe the changes to the graph of the Taylor polynomial as n increases. The approimation to the left of a = 0.3 might appear to be better. The approimation is accurate on a symmetric interval about a = 0.3. However, the arc length of the graph of y ln to the left of a = 0.3 is greater than for a similar interval to the right of a = 0.3. This leads to the graphical suggestion that the approimation is better to the left of a = Teas Instruments Incorporated 4 education.ti.com
5 Move to page In this eample, the graph of y sin( ) is dotted and the graph of the Taylor polynomial of degree n at a is solid. Use the slider arrows to change the degree, n, or the value of a. a. For a = 0 and n =, describe the graph of the Taylor polynomial. Find the Taylor polynomial and describe the approimation for sin for close to 0. Answer: The graph of the Taylor polynomial is a straight line through the origin with slope. The Taylor polynomial is taylorf( ). For close to 0, this suggests sin. b. For a = 0, consider the graph of the Taylor polynomials as n increases. Eplain why the graph of the Taylor polynomials for n = and for n = 2 are identical, and for n = 3 and n = 4, etc. Answer: For i even, f () i ( ) sin and f () i (0) 0. Therefore, there are no terms of even degree in any Taylor polynomial for y sin. The Taylor polynomials for n = 3 and n = 4, for eample, are identical. c. For each value of a and n, describe the accuracy of the Taylor approimation about the point = a. Answer: The Taylor polynomial appears to be accurate on a symmetric interval about the point = a. Move to page In this eample, the graph of y cos is dotted and the graph of the Taylor polynomial of degree n at a is solid. Use the slider arrows to change the degree, n, or the value of a. a. For a = 0 and n =, describe the graph of the Taylor polynomial. Find the Taylor polynomial and eplain why the slope of this linear approimation is 0. Answer: The graph of the Taylor polynomial is a horizontal line through the point (0, ). The Taylor polynomial is taylorf( ). f ( ) sin and f (0) 0. Therefore, the coefficient on the linear term is 0; the slope of this linear approimation is 0. b. For a = 0, consider the graph of the Taylor polynomials as n increases. Eplain why the graph of the Taylor polynomials for n = 0 and for n = are identical, and for n = 2 and n = 3, etc. i i Answer: For i odd, f () ( ) sin and f () (0) 0. Therefore, there are no terms of odd degree in any Taylor polynomial for y cos. The Taylor polynomials for n = 4 and n = 5, for eample, are identical. 20 Teas Instruments Incorporated 5 education.ti.com
6 Move to page In this eample, the graph of y is dotted and the graph of the Taylor polynomial of degree n at a is solid. Use the slider arrows to change the degree, n, or the value of a. a. For a = 0, consider various Taylor polynomials of degree n. Eplain why there is no graph of the Taylor polynomial to the right of =. Answer: The function f( ) has a discontinuity at =. The graph of the Taylor polynomial cannot etend beyond this discontinuity. b. Consider the graph of the Taylor polynomial for a = 0 and n = 7. Eplain the accuracy of this Taylor polynomial. Why does the Taylor polynomial appear to be a much better approimation to the right of a = 0than to the left? Answer: The arc length of the graph of y is greater for 0 than for 0. Therefore, the Taylor polynomial appears to be a better approimation to the right of a = 0. The approimation is accurate on a symmetric interval about a = 0. c. Eplain how to obtain the graph of a Taylor polynomial that can be used to approimate the portion of the graph of y f( ) to the right of =. Answer: Select a value a >. This will produce a Taylor polynomial that can be used to approimate the portion of the graph of y f( ) to the right of =. Wrap Up Upon completion of this activity, the teacher should ensure that students understand: How the accuracy of a Taylor polynomial is associated with the degree of the Taylor polynomial and the value a. A Taylor polynomial is accurate on a symmetric interval about = a. How a point of discontinuity affects a Taylor polynomial. The relationship between the ith derivative of the function and the ith derivative of the corresponding Taylor polynomial. At the end of this activity, you might consider asking students to graph other common functions and the associated Taylor polynomials. For eample, consider f( ) tan, 2 y e, and y e sin. 20 Teas Instruments Incorporated 6 education.ti.com
Polynomials Factors, Roots, and Zeros TEACHER NOTES MATH NSPIRED
Math Objectives Students will discover that the zeros of the linear factors are the zeros of the polynomial function. Students will discover that the real zeros of a polynomial function are the zeros of
More informationEnd Behavior of Polynomial Functions TEACHER NOTES MATH NSPIRED
Math Objectives Students will recognize the similarities and differences among power and polynomial functions of various degrees. Students will describe the effects of changes in the leading coefficient
More informationWhat Is a Solution to a System?
Math Objectives Students will understand what it means for an ordered pair to be a solution to a linear equation. Students will understand what it means for an ordered pair to be a solution to a system
More informationMultiplying Inequalities by Negative Numbers
Math Objectives Students will relate the inequality symbol to the location of points on a number line. Students will recognize, moreover, that the value of a number depends on its position on the number
More informationFinite Differences TEACHER NOTES MATH NSPIRED
Math Objectives Students will be able to recognize that the first set of finite differences for a linear function will be constant. Students will be able to recognize that the second set of finite differences
More informationStandard Error & Sample Means
Math Objectives Students will recognize that samples have smaller variability than the population. Students will recognize that the variability in samples is a function of sample size, n, and that the
More informationArea Measures and Right Triangles
Math Objectives Students will open or create a TI-Nspire document with a right triangle that has equilateral triangles on its sides. The areas of the equilateral triangles are measured and displayed on
More informationTEACHER NOTES MATH NSPIRED
Math Objectives Students will understand how various data transformations can be used to achieve a linear relationship and, consequently, how to use the methods of linear regression to obtain a line of
More informationRose Curve / G. TEACHER NOTES MATH NSPIRED: PRECALCULUS. Math Objectives. Vocabulary. About the Lesson. TI-Nspire Navigator TM
Math Objectives Students will understand the role of the values of a and n in the equation r = asin(nθ). Students will be able to predict the number of petals and their length by examining the polar equation.
More informationEpsilon-Delta Window Challenge TEACHER NOTES MATH NSPIRED
Math Objectives Students will interpret the formal definition (epsilon-delta) of limit in terms of graphing window dimensions. Students will use this interpretation to make judgments of whether a limit
More informationTEACHER NOTES MATH NSPIRED
Math Objectives Students will learn that for a continuous non-negative function f, one interpretation of the definite integral f ( x ) dx is the area of a the region bounded above by the graph of y = f(x),
More informationFirst Derivative Test TEACHER NOTES
Math Objectives Students will relate the first derivative of a function to its critical s and identify which of these critical s are local extrema. Students will visualize why the first derivative test
More informationCrossing the Asymptote
Math Objectives Students will test whether the graph of a given rational function crosses its horizontal asymptote. Students will examine the relationship among the coefficients of the polynomials in the
More informationFrom Rumor to Chaos TEACHER NOTES MATH NSPIRED. Math Objectives. Vocabulary. About the Lesson. TI-Nspire Navigator System
Math Objectives Students will model the spread of a rumor using a recursive sequence. Students will understand that variations in the recursive sequence modeling the spread of a rumor or a disease can
More informationSD: How Far is Typical? TEACHER NOTES
Math Objectives Students will identify the standard deviation as a measure of spread about the mean of a distribution and loosely interpret it as an "average" or typical distance from the mean. Students
More informationSum of an Infinite Geometric Series
Math Objectives Students will understand how a unit square can be divided into an infinite number of pieces. Students will understand a justification for the following theorem: The sum of the infinite
More informationTEACHER NOTES MATH NSPIRED
Math Objectives Students will solve linear equations in one variable graphically and algebraically. Students will explore what it means for an equation to be balanced both graphically and algebraically.
More informationFunction Notation TEACHER NOTES MATH NSPIRED. Math Objectives. Vocabulary. About the Lesson. TI-Nspire Navigator System. Activity Materials
Math Objectives Students will understand function notation and the distinction between an input value x, a function f, and a function output f(x) Students will determine the rule for a function machine
More informationTEACHER NOTES SCIENCE NSPIRED
Science Objectives Students will explore the force two charges exert on each other. Students will observe how the force depends upon the magnitude of the two charges and the distance separating them. Students
More informationIt s Just a Lunar Phase
Science Objectives Students will identify patterns in data associated with the lunar phases. Students will describe how the relative positions of the Earth, the Moon, and the Sun cause lunar phases. Students
More informationInfluencing Regression
Math Objectives Students will recognize that one point can influence the correlation coefficient and the least-squares regression line. Students will differentiate between an outlier and an influential
More informationTEACHER NOTES MIDDLE GRADES SCIENCE NSPIRED
Science Objectives Students will describe the three states of matter. Students will analyze changes of state graphically. Students will ascertain the melting and boiling points of different substances.
More informationCommon Core State Standards for Activity 14. Lesson Postal Service Lesson 14-1 Polynomials PLAN TEACH
Postal Service Lesson 1-1 Polynomials Learning Targets: Write a third-degree equation that represents a real-world situation. Graph a portion of this equation and evaluate the meaning of a relative maimum.
More informationSweating Alcohol TEACHER NOTES SCIENCE NSPIRED. Science Objectives. Math Objectives. Materials Needed. Vocabulary.
Science Objectives Students will collect data for the cooling rates of water and isopropyl alcohol. Students will compare and contrast the cooling rate data, both graphically and numerically. Students
More informationTImath.com Calculus. Topic: Techniques of Integration Derive the formula for integration by parts and use it to compute integrals
Integration by Parts ID: 985 Time required 45 minutes Activity Overview In previous activities, students have eplored the differential calculus through investigations of the methods of first principles,
More informationTEACHER NOTES SCIENCE NSPIRED
Science Objectives Students will determine the relationship between mass and volume. Students will mathematically describe the relationship between mass and volume. Students will relate the slope of a
More informationElectric Field Hockey
Science Objectives Students will describe an electric field and electric field lines. Students will describe what happens when two like charges interact and when two unlike charges interact. Students will
More informationAP Calculus BC Prerequisite Knowledge
AP Calculus BC Prerequisite Knowledge Please review these ideas over the summer as they come up during our class and we will not be reviewing them during class. Also, I feel free to quiz you at any time
More informationPre-Calculus Module 4
Pre-Calculus Module 4 4 th Nine Weeks Table of Contents Precalculus Module 4 Unit 9 Rational Functions Rational Functions with Removable Discontinuities (1 5) End Behavior of Rational Functions (6) Rational
More informationA Magnetic Attraction
Science Objectives Students will map and describe the magnetic field around a permanent magnet or electromagnet. Students will determine the direction and strength of the force a magnet exerts on a current-carrying
More informationFriction: Your Friend or Your Enemy?
Science Objectives Students will determine what factors affect the friction between two surfaces. Students will relate the forces needed to drag different shoes across a table at a constant speed. Students
More informationSpecial Right Triangles
Math Objectives Students will identify the relationship between the lengths of the shorter and longer legs in a 30-60 -90 right triangle. Students will determine the relationship between the shorter leg
More informationDiscovering e. Mathematical Content: Euler s number Differentiation of exponential functions Integral of 1/x Sequences & series
Discovering e Mathematical Content: Euler s number Differentiation of exponential functions Integral of 1/x Sequences & series Technical TI-Nspire Skills: Manipulation of geometric constructions Use of
More informationTImath.com Algebra 1. Trigonometric Ratios
Algebra 1 Trigonometric Ratios ID: 10276 Time required 60 minutes Activity Overview In this activity, students discover the trigonometric ratios through measuring the side lengths of similar triangles
More informationTEACHER NOTES SCIENCE NSPIRED
Science Objectives Students will investigate the effects of mass, spring constant, and initial displacement on the motion of an object oscillating on a spring. Students will observe the relationships between
More informationAccel Alg E. L. E. Notes Solving Quadratic Equations. Warm-up
Accel Alg E. L. E. Notes Solving Quadratic Equations Warm-up Solve for x. Factor. 1. 12x 36 = 0 2. x 2 8x Factor. Factor. 3. 2x 2 + 5x 7 4. x 2 121 Solving Quadratic Equations Methods: (1. By Inspection)
More information7.4. The Primary Trigonometric Ratios. LEARN ABOUT the Math. Connecting an angle to the ratios of the sides in a right triangle. Tip.
The Primary Trigonometric Ratios GOL Determine the values of the sine, cosine, and tangent ratios for a specific acute angle in a right triangle. LERN OUT the Math Nadia wants to know the slope of a ski
More informationBuilding Concepts: Solving Systems of Equations Algebraically
Lesson Overview In this TI-Nspire lesson, students will investigate pathways for solving systems of linear equations algebraically. There are many effective solution pathways for a system of two linear
More informationModule 2, Section 2 Solving Equations
Principles of Mathematics Section, Introduction 03 Introduction Module, Section Solving Equations In this section, you will learn to solve quadratic equations graphically, by factoring, and by applying
More informationDISCOVERING CALCULUS WITH THE TI- NSPIRE CAS CALCULATOR
John Carroll University Carroll Collected Masters Essays Theses, Essays, and Senior Honors Projects Summer 017 DISCOVERING CALCULUS WITH THE TI- NSPIRE CAS CALCULATOR Meghan A. Nielsen John Carroll University,
More informationM&M Exponentials Exponential Function
M&M Exponentials Exponential Function Teacher Guide Activity Overview In M&M Exponentials students will experiment with growth and decay functions. Students will also graph their experimental data and
More information3.2 Logarithmic Functions and Their Graphs
96 Chapter 3 Eponential and Logarithmic Functions 3.2 Logarithmic Functions and Their Graphs Logarithmic Functions In Section.6, you studied the concept of an inverse function. There, you learned that
More informationTEACHER NOTES SCIENCE NSPIRED
Science Objectives Students will transfer electrons from a metal to a nonmetal. Students will relate this transfer to the formation of positive and negative ions. Student will learn how to write formulas
More information2.6 Solving Inequalities Algebraically and Graphically
7_006.qp //07 8:0 AM Page 9 Section.6 Solving Inequalities Algebraically and Graphically 9.6 Solving Inequalities Algebraically and Graphically Properties of Inequalities Simple inequalities were reviewed
More informationReview: Properties of Exponents (Allow students to come up with these on their own.) m n m n. a a a. n n n m. a a a. a b a
Algebra II Notes Unit Si: Polynomials Syllabus Objectives: 6. The student will simplify polynomial epressions. Review: Properties of Eponents (Allow students to come up with these on their own.) Let a
More informationLimits and Their Properties
Chapter 1 Limits and Their Properties Course Number Section 1.1 A Preview of Calculus Objective: In this lesson you learned how calculus compares with precalculus. I. What is Calculus? (Pages 42 44) Calculus
More informationKinetics Activity 1 Objective
Grade level: 9-12 Subject: Chemistry Time required: 45 to 90 minutes Kinetic by Todd Morstein Study of Chemical Rates of Reaction An introductory lesson on determining the differential rate law and rate
More informationAvon High School Name AP Calculus AB Summer Review Packet Score Period
Avon High School Name AP Calculus AB Summer Review Packet Score Period f 4, find:.) If a.) f 4 f 4 b.) Topic A: Functions f c.) f h f h 4 V r r a.) V 4.) If, find: b.) V r V r c.) V r V r.) If f and g
More informationName Student Activity
Open the TI-Nspire document From_Rumor_to_Chaos.tns. In this investigation, you will model the spread of a rumor using a recursive sequence to see how some variations of this model can lead to chaos. Terry
More informationPrecalculus Notes: Functions. Skill: Solve problems using the algebra of functions.
Skill: Solve problems using the algebra of functions. Modeling a Function: Numerical (data table) Algebraic (equation) Graphical Using Numerical Values: Look for a common difference. If the first difference
More informationMath Analysis/Honors Math Analysis Summer Assignment
Math Analysis/Honors Math Analysis Summer Assignment To be successful in Math Analysis or Honors Math Analysis, a full understanding of the topics listed below is required prior to the school year. To
More informationAPPM 1360 Final Exam Spring 2016
APPM 36 Final Eam Spring 6. 8 points) State whether each of the following quantities converge or diverge. Eplain your reasoning. a) The sequence a, a, a 3,... where a n ln8n) lnn + ) n!) b) ln d c) arctan
More information3.3.1 Linear functions yet again and dot product In 2D, a homogenous linear scalar function takes the general form:
3.3 Gradient Vector and Jacobian Matri 3 3.3 Gradient Vector and Jacobian Matri Overview: Differentiable functions have a local linear approimation. Near a given point, local changes are determined by
More informationSolving Stoichiometry Problems
Science Objectives Students will write ratios from balanced chemical equations. Students will use -to-mass conversion to determine mass of reactants and products iven the number of s. Students will calculate
More informationEssential Question How can you verify a trigonometric identity?
9.7 Using Trigonometric Identities Essential Question How can you verify a trigonometric identity? Writing a Trigonometric Identity Work with a partner. In the figure, the point (, y) is on a circle of
More informationTIphysics.com. Physics. Bell Ringer: Determining the Relationship Between Displacement, Velocity, and Acceleration ID: 13308
Bell Ringer: Determining the Relationship Between Displacement, Velocity, and Acceleration ID: 13308 Time required 15 minutes Topic: Kinematics Construct and interpret graphs of displacement, velocity,
More information11.11 Applications of Taylor Polynomials. Copyright Cengage Learning. All rights reserved.
11.11 Applications of Taylor Polynomials Copyright Cengage Learning. All rights reserved. Approximating Functions by Polynomials 2 Approximating Functions by Polynomials Suppose that f(x) is equal to the
More informationRadical and Rational Functions
Radical and Rational Functions 5 015 College Board. All rights reserved. Unit Overview In this unit, you will etend your study of functions to radical, rational, and inverse functions. You will graph radical
More information9.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED LESSON
CONDENSED LESSON 9.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations solve
More informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus.1 Worksheet Day 1 All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. The only way to guarantee the eistence of a it is to algebraically prove
More informationPhysics. Simple Harmonic Motion ID: 9461
Simple Harmonic Motion ID: 9461 By Peter Fox Time required 45 minutes Activity Overview In this activity, students collect data on the motion of a simple pendulum. They then graph the acceleration of the
More informationTIphysics.com. Physics. Pendulum Explorations ID: By Irina Lyublinskaya
Pendulum Explorations ID: 17 By Irina Lyublinskaya Time required 90 minutes Topic: Circular and Simple Harmonic Motion Explore what factors affect the period of pendulum oscillations. Measure the period
More informationLearning Targets: Standard Form: Quadratic Function. Parabola. Vertex Max/Min. x-coordinate of vertex Axis of symmetry. y-intercept.
Name: Hour: Algebra A Lesson:.1 Graphing Quadratic Functions Learning Targets: Term Picture/Formula In your own words: Quadratic Function Standard Form: Parabola Verte Ma/Min -coordinate of verte Ais of
More informationTEACHER NOTES TIMATH.COM: ALGEBRA 2 ID: and discuss the shortcomings of this
Activity Overview In this activity students explore models for the elliptical orbit of Jupiter. Problem 1 reviews the geometric definition of an ellipse as students calculate for a and b from the perihelion
More informationIntroduction to Rational Functions
Introduction to Rational Functions The net class of functions that we will investigate is the rational functions. We will eplore the following ideas: Definition of rational function. The basic (untransformed)
More information7.1 Indefinite Integrals Calculus
7.1 Indefinite Integrals Calculus Learning Objectives A student will be able to: Find antiderivatives of functions. Represent antiderivatives. Interpret the constant of integration graphically. Solve differential
More informationRoberto s Notes on Differential Calculus Chapter 1: Limits and continuity Section 7. Discontinuities. is the tool to use,
Roberto s Notes on Differential Calculus Chapter 1: Limits and continuity Section 7 Discontinuities What you need to know already: The concept and definition of continuity. What you can learn here: The
More informationAP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals
AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals 8. Basic Integration Rules In this section we will review various integration strategies. Strategies: I. Separate
More informationScience Friction! TEACHER NOTES. Science Objectives. Vocabulary. About the Lesson. TI-Nspire Navigator. Activity Materials
Science Objectives Students will investigate the role of friction in the motion of a hero running across a concrete surface. Students will learn about Newton s second and third laws of motion and how they
More informationy = x 2 b) Graph c) Summary x-intercepts
A 8-5 Name BDFM? Why? AStd5c: I can graph quadratic functions. Investigate the parabola. Circle the -intercepts, draw the line of symmetry, put a star net to the y-intercept, and put a v net to the verte.
More informationRegion 16 Board of Education. Precalculus Curriculum
Region 16 Board of Education Precalculus Curriculum 2008 1 Course Description This course offers students an opportunity to explore a variety of concepts designed to prepare them to go on to study calculus.
More informationMORE CURVE SKETCHING
Mathematics Revision Guides More Curve Sketching Page of 3 MK HOME TUITION Mathematics Revision Guides Level: AS / A Level MEI OCR MEI: C4 MORE CURVE SKETCHING Version : 5 Date: 05--007 Mathematics Revision
More informationColumbus City Schools High School CCSS Mathematics III - High School PARRC Model Content Frameworks Mathematics - Core Standards And Math Practices
A Correlation of III Common Core To the CCSS III - - Core Standards And s A Correlation of - III Common Core, to the CCSS III - - Core Standards and s Introduction This document demonstrates how - III
More informationINVESTIGATE the Math
. Graphs of Reciprocal Functions YOU WILL NEED graph paper coloured pencils or pens graphing calculator or graphing software f() = GOAL Sketch the graphs of reciprocals of linear and quadratic functions.
More informationTower of PISA. Standards Addressed
Tower of PISA Standards Addressed. The Standards for Mathematical Practice, especially:. Make sense of problems and persevere in solving them and. Reason abstractly and quantitatively.. 8.G.B.5: Apply
More informationMath Honors Calculus I Final Examination, Fall Semester, 2013
Math 2 - Honors Calculus I Final Eamination, Fall Semester, 2 Time Allowed: 2.5 Hours Total Marks:. (2 Marks) Find the following: ( (a) 2 ) sin 2. (b) + (ln 2)/(+ln ). (c) The 2-th Taylor polynomial centered
More informationMultiplying and Dividing Rational Expressions
6.3 Multiplying and Dividing Rational Epressions Essential Question How can you determine the ecluded values in a product or quotient of two rational epressions? You can multiply and divide rational epressions
More informationthrough any three given points if and only if these points are not collinear.
Discover Parabola Time required 45 minutes Teaching Goals: 1. Students verify that a unique parabola with the equation y = ax + bx+ c, a 0, exists through any three given points if and only if these points
More informationEpsilon-Delta Window Challenge Name Student Activity
Open the TI-Nspire document Epsilon-Delta.tns. In this activity, you will get to visualize what the formal definition of limit means in terms of a graphing window challenge. Informally, lim f( x) Lmeans
More informationSecondary Math 3 Honors Unit 10: Functions Name:
Secondary Math 3 Honors Unit 10: Functions Name: Parent Functions As you continue to study mathematics, you will find that the following functions will come up again and again. Please use the following
More informationWelcome to Advanced Placement Calculus!! Summer Math
Welcome to Advanced Placement Calculus!! Summer Math - 017 As Advanced placement students, your first assignment for the 017-018 school year is to come to class the very first day in top mathematical form.
More informationUsing the TI-92+ : examples
Using the TI-9+ : eamples Michel Beaudin École de technologie supérieure 1100, rue Notre-Dame Ouest Montréal (Québec) Canada, H3C 1K3 mbeaudin@seg.etsmtl.ca 1- Introduction We incorporated the use of the
More informationChapter 3. Exponential and Logarithmic Functions. Selected Applications
Chapter 3 Eponential and Logarithmic Functions 3. Eponential Functions and Their Graphs 3.2 Logarithmic Functions and Their Graphs 3.3 Properties of Logarithms 3.4 Solving Eponential and Logarithmic Equations
More information3.5 Continuity of a Function One Sided Continuity Intermediate Value Theorem... 23
Chapter 3 Limit and Continuity Contents 3. Definition of Limit 3 3.2 Basic Limit Theorems 8 3.3 One sided Limit 4 3.4 Infinite Limit, Limit at infinity and Asymptotes 5 3.4. Infinite Limit and Vertical
More informationarb where a A, b B and we say a is related to b. Howdowewritea is not related to b? 2Rw 1Ro A B = {(a, b) a A, b B}
Functions Functions play an important role in mathematics as well as computer science. A function is a special type of relation. So what s a relation? A relation, R, from set A to set B is defined as arb
More informationHorizontal asymptotes
Roberto s Notes on Differential Calculus Chapter : Limits and continuity Section 5 Limits at infinity and Horizontal asymptotes What you need to know already: The concept, notation and terminology of its.
More informationExample 1: What do you know about the graph of the function
Section 1.5 Analyzing of Functions In this section, we ll look briefly at four types of functions: polynomial functions, rational functions, eponential functions and logarithmic functions. Eample 1: What
More informationHorizontal asymptotes
Roberto s Notes on Differential Calculus Chapter 1: Limits and continuity Section 5 Limits at infinity and Horizontal asymptotes What you need to know already: The concept, notation and terminology of
More informationSYSTEMS OF THREE EQUATIONS
SYSTEMS OF THREE EQUATIONS 11.2.1 11.2.4 This section begins with students using technology to eplore graphing in three dimensions. By using strategies that they used for graphing in two dimensions, students
More informationAP Calculus BC Syllabus
AP Calculus BC Syllabus Course Overview and Philosophy This course is designed to be the equivalent of a college-level course in single variable calculus. The primary textbook is Calculus, 7 th edition,
More informationCalculus I Practice Test Problems for Chapter 2 Page 1 of 7
Calculus I Practice Test Problems for Chapter Page of 7 This is a set of practice test problems for Chapter This is in no way an inclusive set of problems there can be other types of problems on the actual
More informationMaths A Level Summer Assignment & Transition Work
Maths A Level Summer Assignment & Transition Work The summer assignment element should take no longer than hours to complete. Your summer assignment for each course must be submitted in the relevant first
More informationSet 3: Limits of functions:
Set 3: Limits of functions: A. The intuitive approach (.): 1. Watch the video at: https://www.khanacademy.org/math/differential-calculus/it-basics-dc/formal-definition-of-its-dc/v/itintuition-review. 3.
More informationX ; this can also be expressed as
The Geometric Mean ID: 9466 TImath.com Time required 30 minutes Activity Overview In this activity, students will establish that several triangles are similar and then determine that the altitude to the
More informationTIphysics.com. Physics. Collisions in One Dimension ID: 8879
Collisions in One Dimension ID: 8879 By Charles W. Eaker Time required 45 minutes Topic: Momentum and Collisions Compare kinetic energy and momentum. Classify collisions as elastic or inelastic. Compare
More informationTEACHER NOTES MIDDLE GRADES SCIENCE NSPIRED
Science Objectives Students will investigate and differentiate between planets, moons, and asteroids. Students will understand how planets are classified. Vocabulary asteroid planet moon gravity orbital
More informationSolving Stoichiometry Problems
Open the TI-Nspire document Solving_Stoichiometry_Problems.tns In this lesson you will use simulations of three combustion reactions to develop skills necessary to solve stoichiometric problems. Move to
More informationSummer AP Assignment Coversheet Falls Church High School
Summer AP Assignment Coversheet Falls Church High School Course: AP Calculus AB Teacher Name/s: Veronica Moldoveanu, Ethan Batterman Assignment Title: AP Calculus AB Summer Packet Assignment Summary/Purpose:
More informationUnit 6 Quadratic Relations of the Form y = ax 2 + bx + c
Unit 6 Quadratic Relations of the Form y = ax 2 + bx + c Lesson Outline BIG PICTURE Students will: manipulate algebraic expressions, as needed to understand quadratic relations; identify characteristics
More information